Question: What is the extraneous solution to these equations? $\dfrac{x^2 + 75}{x - 9} = \dfrac{156}{x - 9}$
Answer: Multiply both sides by $x - 9$ $ \dfrac{x^2 + 75}{x - 9} (x - 9) = \dfrac{156}{x - 9} (x - 9)$ $ x^2 + 75 = 156$ Subtract $156$ from both sides: $ x^2 + 75 - (156) = 156 - (156)$ $ x^2 + 75 - 156 = 0$ $ x^2 - 81 = 0$ Factor the expression: $ (x + 9)(x - 9) = 0$ Therefore $x = -9$ or $x = 9$ At $x = 9$ , the denominator of the original expression is 0. Since the expression is undefined at $x = 9$, it is an extraneous solution.